Surfaces expanding by the power of the Gauss curvature flow

Li, Qi-Rui

doi:10.1090/S0002-9939-2010-10431-8

Surfaces expanding by the power of the Gauss curvature flow
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by Qi-Rui Li

Proc. Amer. Math. Soc. 138 (2010), 4089-4102

DOI: https://doi.org/10.1090/S0002-9939-2010-10431-8

Published electronically: June 16, 2010

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Abstract:

In this paper, we describe the flow of 2-surfaces in $\mathbb {R}^{3}$ for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity $K^{-\alpha }$, when $\frac {1}{2}<\alpha \leq 1$, converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for $\alpha \in (0,\frac {1}{2}]$ by Urbas and Huisken and for $\alpha =1$ by Schnürer.

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